add solver for dynamic linear economies as LQ problem

[mmcky]

This PR adds DLE object for solving dynamic linear economic problems by converting to LQ problem. This code has been contributed by Sebastian Graves.

Adjustments required for inclusion in QuantEcon.py:

• update docstring to conform to style guide. Need to document Parameters in similar fashion to https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/robustlq.py

• review inclusion of pylab. I would prefer to use the core libraries numpy, matplotlib and scipy

• add tests

[mmcky]

is this simply importing a numpy array?

[oyamad]

• Comply with PEP8.

[oyamad]

Can this be replaced with quantecon.nullspace? Then the import of Matrix from sympy would be removed.

[mmcky]

@oyamad I am in favour of this. Can you let me know what I need to change?

[mmcky]

self.zz = qe.nullspace(A0)?

[oyamad]

@mmcky What I had in mind is

zx = np.eye(self.A0.shape[0]) - self.A0
self.zz = qe.nullspace(zx)

Maybe the following is better?

zx = -self.A0
zx[np.diag_indices_from(zx)] += 1
self.zz = qe.nullspace(zx)
self.zz /= self.zz[nnc]

But please check whether this gives the same shape of self.zz as the original code.

[mmcky]

thanks @oyamad. I will make the update and then confirm.

[coveralls]

Coverage decreased (-0.8%) to 94.389% when pulling 54be06a on add-dle into 9a47420 on master.

[mmcky]

@thomassargent30 this will default to using doubling as the method for computing. Should we allow the pass through of the method of computation to be doubling and qz?

[mmcky]

@oyamad the nullspace method attached to Sympy Matrix Returns list of vectors (Matrix objects) that span nullspace of self. The quantecon nullspace returns an approximate basis for the nullspace of A. I have reverted to using the sympy Matrix object for now and will open an issue to remove.

[oyamad]

@mmcky Do you have an example that yields such a difference?

[mmcky]

@oyamad

Given the input matrix

source = np.array([[ 0.1 ,  0.  , -0.5 , -0.1 ,  0.  ],
       [ 0.  ,  0.05,  0.  ,  0.  ,  0.  ],
       [ 0.  ,  0.  ,  0.  ,  0.  ,  0.  ],
       [ 0.  ,  0.  ,  0.  ,  0.2 ,  0.  ],
       [ 0.  ,  0.  ,  0.  ,  0.  ,  0.5 ]])

using null space method from the Matrix object in Simps

from sympy import Matrix
Matrix(source).nullspace()

yields

[Matrix([
 [5.0],
 [  0],
 [  1],
 [  0],
 [  0]])]

while null space method in quantecon

from quantecon import nullspace
nullspace(source)

yields

array([[ 9.80580676e-01],
       [-2.77555756e-17],
       [ 1.96116135e-01],
       [ 1.38777878e-17],
       [ 0.00000000e+00]])

[oyamad]

What is the problem exactly?
The approximation looks good:

zz = qe.nullspace(source)
nnc = 2
zz /= zz[nnc]
zz

array([[ 5.00000000e+00],
       [-1.41526222e-16],
       [ 1.00000000e+00],
       [ 7.07631108e-17],
       [ 0.00000000e+00]])

[mmcky]

ah thanks @oyamad I didn't divide by the constant. I will update the dle class now.

[mmcky]

@oyamad are you seeing Travis in your browser? It appears to be running on Travis but not showing status on this github PR for me.

[mmcky]

thanks @oyamad for picking up my error. This has now been migrated to using the quantecon method and testing has been updating on a known problem.